.TH std::legendre,std::legendref,std::legendrel 3 "2024.06.10" "http://cppreference.com" "C++ Standard Libary"
.SH NAME
std::legendre,std::legendref,std::legendrel \- std::legendre,std::legendref,std::legendrel

.SH Synopsis
   double      legendre( unsigned int n, double x );

   double      legendre( unsigned int n, float x );
   double      legendre( unsigned int n, long double x );  \fB(1)\fP
   float       legendref( unsigned int n, float x );

   long double legendrel( unsigned int n, long double x );
   double      legendre( unsigned int n, IntegralType x ); \fB(2)\fP

   1) Computes the unassociated Legendre polynomials of the degree n and argument x.
   2) A set of overloads or a function template accepting an argument of any integral
   type. Equivalent to \fB(1)\fP after casting the argument to double.

   As all special functions, legendre is only guaranteed to be available in <cmath> if
   __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least
   201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any
   standard library headers.

.SH Parameters

   n - the degree of the polynomial
   x - the argument, a value of a floating-point or integral type

.SH Return value

   If no errors occur, value of the order-n unassociated Legendre polynomial of x, that
   is

   1
   2n
   n!

   dn
   dxn

   (x2
   - 1)n
   , is returned.

.SH Error handling

   Errors may be reported as specified in math_errhandling.

     * If the argument is NaN, NaN is returned and domain error is not reported.
     * The function is not required to be defined for |x| > 1.
     * If n is greater or equal than 128, the behavior is implementation-defined.

.SH Notes

   Implementations that do not support TR 29124 but support TR 19768, provide this
   function in the header tr1/cmath and namespace std::tr1.

   An implementation of this function is also available in boost.math.

   The first few Legendre polynomials are:

     * legendre(0, x) = 1.
     * legendre(1, x) = x.
     * legendre(2, x) =

       1
       2

       (3x2
       - 1).
     * legendre(3, x) =

       1
       2

       (5x3
       - 3x).
     * legendre(4, x) =

       1
       8

       (35x4
       - 30x2
       + 3).

.SH Example

   (works as shown with gcc 6.0)


// Run this code

 #define __STDCPP_WANT_MATH_SPEC_FUNCS__ 1
 #include <cmath>
 #include <iostream>

 double P3(double x)
 {
     return 0.5 * (5 * std::pow(x, 3) - 3 * x);
 }

 double P4(double x)
 {
     return 0.125 * (35 * std::pow(x, 4) - 30 * x * x + 3);
 }

 int main()
 {
     // spot-checks
     std::cout << std::legendre(3, 0.25) << '=' << P3(0.25) << '\\n'
               << std::legendre(4, 0.25) << '=' << P4(0.25) << '\\n';
 }

.SH Output:

 -0.335938=-0.335938
 0.157715=0.157715

.SH See also

   laguerre  Laguerre polynomials
   laguerref \fI(function)\fP
   laguerrel
   hermite   Hermite polynomials
   hermitef  \fI(function)\fP
   hermitel

.SH External links

   Weisstein, Eric W. "Legendre Polynomial." From MathWorld — A Wolfram Web Resource.
